## How to Partially Solve a System of Equations...

I'm attempting to find the eigenvectors of a matrix without using the eigenvector function.

The matrix in question is a covariance matrix:

`XCov:=Matrix([[4048/5, -817/5, -122/5], [-817/5, 921/10, -1999/10], [-122/5, -1999/10, 8341/10]]);`

I've already found the eigenvalues by solving for lambda:

``` det := Determinant(XCov-lambda*IdentityMatrix(3));   lambda := solve(det=0.0, lambda);```

(Yes I'm reusing the eigenvalue variable for the set of eigenvalues once they've been found😏)

Anyway, I've now set up the first eigenvector I want to find as:
`e1 := Vector([e11,e12,e13]);`

Now, the equation to find this first eigenvector is XCov . e1 = lambda . e1
I first tried putting whats on the left in a variable called eigscale(what the vector is translated to by the matrix):

`eigscale := Multiply(XCov,e1);`
Which returns a vector:
eigscale = [(4048/5)*e11-(817/5)*e12-(122/5)*e13,
-(817/5)*e11+(921/10)*e12-(1999/10)*e13,
-(122/5)*e11-(1999/10)*e12+(8341/10)*e13]

Each component of this vector must equate to the corresponding component in the right vector:

lambda*e1 = [7.943520930*e11, 7.943520930*e12, 7.943520930*e13]

At first I tried setting these vectors equal to each other and using a solve but of course it didnt like the equations being in a vector format. So I then seperated out each equation and gave the solve function a system of equations as it expects:

`solve(eigscale = lambda*e1, eigscale = lambda*e1, eigscale = lambda*e1, [e11,e12,e13]);`

But again, solve fails to solve them. The reason this time(I believe) is because it can't find an exact value for e11, e12 & e13.
When solving for an eigenvector we get
`e11 = e11`,
`e12 = Ae11`,
`e13 = Be11 + Ce12`

I was wondering if there was a way to do a partial solve to find the components in terms of each other?

Failing that, I'm aware I can do it manually through row operations but I believe that would require changing the format so that each equation is a component of a single vector:
`eigsolve := Vector([eigscale = lambda*e1, eigscale = lambda*e1, eigscale = lambda*e1]);`

Since row operations cannot be performed on a equation of vectors (again, I believe).

Help appreciated!

## How to find out intervals of a variables where fun...

Hello Everyone,

I have been working on a multi variable expression. I would like to have the intervals where the function is monotonically increasing. I am trying to study any available method for multivariable expressions.

I came across various papers and sites which are explicitly mentioned single variable equations. finding out the critical points and studying the sign of the first derivative. Same cannot be applied for the multi variable expression.

https://www.math24.net/monotonic-functions/

above link explain for single variable functions. I would be grateful if someone could explain me a method or idea  which helps me out in solving for multivariable functions

## Fitting system of odes by Optimization...

```restart;
Digits := 25;

# Setup.
de := diff( x(t), t ) = a * y(t), diff( y(t), t ) = b * x(t);
ic := x(0) = 1, y(0) = 0;
sol := dsolve( { de, ic } ):

# Sample values for specific parameter values.
a0, b0 := 1.0, 0.25:
T := evalf( [ seq( i, i=1..5 ) ] ):
X := [ seq( eval['recurse']( x(t), [ op( sol ), a = a0, b = b0 ] ), t=T ) ]:
Y := [ seq( eval['recurse']( y(t), [ op( sol ), a = a0, b = b0 ] ), t=T ) ]:

# Numerical solution.
sol := dsolve( { de, ic }, 'numeric', 'range'=min(T)..max(T), 'abserr'=1e-15, 'maxfun'=0, 'parameters'=[a,b] ):

# Procedure to return values for specific time and parameter values.
u := proc( t, a, b )
sol( 'parameters' = [ ':-a' = a, ':-b' = b ] ):
try
return eval( [ x( ':-t' ), y( ':-t' ) ], sol( t ) ):
catch:
return 10000:
end try:
end proc:

# Procedure to return objective function.
r := proc( a, b )
local A, t, p, q, x, y:
A := ListTools:-Flatten( [ seq( u( t, a, b ), t=T ) ] -~ zip( (x,y) -> [x,y], X, Y ) ):
return foldl( (p,q) -> p + q^2, 0, op(A) ):
end proc:

# Find parameter values for bet fit.
Optimization:-Minimize( 'r'(a,b), a=-10..10, b=-10..10 );
```

I was provided the code above by Maple support, to fit a model to data using the Optimization package. I have tried to adapt this to a situation where one might only have the X variable data and so the model should only fit to that data.  I did this by deleting the , y( ':-t' ) in the try catch statement and then having the zip statement read as zip( (x) -> [x], X).

When I try to run that code, I get the error:

Error, (in Optimization:-NLPSolve) invalid input: zip uses a 3rd argument, b, which is missing

Anybody have an idea how to solve this?

Jo

## Physics Package: Projectors between disjointed/com...

Hi,

I try to define the action of projectors of two discrete basis onto a general state. This works as expected when I define the projector by myself. However, when using the "Projector" command, I get a not fully simplified result; see below. It seems like there is a confusion with dot/tensor product.  Can somoeone help?

Best,

Henrik   (1)   (2)  (3)  (4)   (5)  (6) ## hidden characters in worksheet...

Two lines that look the same, produce different results. The first lines gives an error message, but the next line that looks the same, does not.

copying and pasting both lines in Notepad reveals the difference:

Determinant*(R1 . B+R2 . B+R3 . B+R4 . A)

Determinant(R1 . B+R2 . B+R3 . B+R4 . A)

It seems that there is a hidden character (the asterisk) in the first line that produces the error.

In the worksheet itself you cannot see the asterisk, but using the arrow keys you can notice that there is another character.

It's hard to debug your code if there are hidden characters. > >  (1)
 > > > > >  (2) ## how to plot a nested for loop ?...

i want to generate grid for a C_D nozzle but i dont know ho to plot it. please help me how to plot.

restart;
alpha := 0;
beta := 2;
Imax := 11;
Jmax := 11;
L := 10;
`&Delta;&zeta;` := L/(Imax-1);

`&Delta;x` := `&Delta;&zeta;`;
`&Delta;&eta;` := 1/(Jmax-1);

printlevel := 2;
for i to Imax do for j to Jmax do x[i] := `&Delta;&zeta;`*(i-1); eta[j] := `&Delta;&eta;`*(j-1); z := (eta[j]-alpha)*ln((beta+1)/(beta-1))/(1-alpha); H[i] := 3+2*tanh(x[i]-6); H1[i] := 0; H2[i] := -H[i]; y[i, j] := -H[i]*(2*beta/(exp(z)+1)-beta-2*alpha)/(2*alpha+1); Y[i, j] := (H1[i]+(-H1[i]*x[i]+H2[i])/L)*eta[j] end do end do;

## simplification of arguments...

If I have an expression like this

f:=ln((1-x)^2*(x+1)^2/((-I*x-I+sqrt(-x^2+1))^2*(I*x+I+sqrt(-x^2+1))^2))

maple has trouble to simplify the argument.

In particular is it possible to apply expand() only to the denominator?

This is meant in general, so if I have many terms with expressions like this (possibly of products with other functions in each term), I want this simplification to be done termwise for the arguments of the functions.

Expanding the fraction doesn't work as in frontend(expand, [f]).

## ANOVA How to execute a procedure?...

Hi,

I am looking to test ANOVA ( Two Way).Maple does not directly gives this possibility.I found a procedure from Mr Tannis but i can t run it? Some ideas?

ThanksANOVA2.mw

## polylog vs. dilog...

This is not a problem per se, but more to understand the background.

restart;

f := polylog(2, -x);

int(f/(x+1), x);

convert(f, dilog);

int(%/(x+1), x)

The integration of the polylog maple is not capable of doing, but after converting to dilog it finds an anti derivative.

That leads to the question, why is dilog as a separate to polylog(2,*) implemented anyway? Why couldn't it all be done with the more general polylog function?

I'm also wondering why maple has difficulties to integrate

int(dilog(x+1)/(x+a),x)

for general a.

## How do I solve an propositions such as this (exist...

How would I go about getting true or false returned on these propositions?
I have tried just about every eval and various syntax methods, but nothing has worked so far.

I know most can easilly be done by hand/thinking, but I'm sure Maple should have a way to do this as well.

∀n∈Z:2n>n+2   ,   ∃n∈Z:2|(3n+1)    ,   ∃k∈Z:∀n∈Z:n=kn   ,   ∃k∈Z:∀n∈Z:2|(n+k)   ,   ∀n∈Z:∀k∈Z:(n>k∨k≥n) ## ImportMatrix Kernel Lost...

Using Maple 2018.2.1, I'm receiving a lost kernel message when importing the attached data file with ImportMatrix. I traced the issue to a "*" symbol at the end of the file but would have expected this to cause an error message (if any error at all) instead of the connection to the kernel to be lost. Is this a bug or am I misunderstanding the usage of ImportMatrix?

test.mw

test2.txt

## Inversion of high dimension matrices...

Hi
How we can obtain inversion of the high-dimensional matrix (for example 600*600) in maple?
(A fast-time command)
thanks

## initial vaule problem...

i want someone hlep me in this worksheet the diff eq of complex i want to sovle it with any numeric method

 > > > >      (1) >  (2)
 > > > > > > > > > > > > > > > > > > > >  (3)
 > > > > > > > > >  (4)
 >  (5)
 >  (6)
 > >   ## Evaluate and print values...

how to evaulate the value of R(z) for different values of z=0 to 1 with an interval of 0.1 and print ten values  in one column

R(z):= 1-cos^2*(Pi*z);

## Fractional-order expansion of a function ...

Hi
I am trying to expand a function f(t) in terms of fractional power series:
for example please see attached file tree.mw > > > > >  